It’s like this other thing except …

One of my complaints about math writing is that definitions are hardly ever subtractive, even if that’s how people think of them.

For example, a monoid is a group except without inverses. But that’s not how you’ll see it defined. Instead you’ll read that it’s a set with an associative binary operation and an identity element. A module is a vector space, except the scalars come from a ring instead of a field. But the definition from scratch is more than I want to write here. Any time you have a sub-widget or a pre-widget or a semi-widget, it’s probably best to define the widget first.

I understand the logical tidiness of saying what a thing is rather than what it is not. But it makes more pedagogical sense to describe the difference between a new concept and the most similar familiar concept. And the nearest familiar concept may have more structure rather than less.

Suppose you wanted to describe where a smaller city is by giving directions from larger, presumably more well known city, but you could only move east. Then instead of saying Ft. Worth is 30 miles west of Dallas, you’d have to say it’s 1,000 miles east of Phoenix.

Writers don’t have to chose between crisp logic and good pedagogy. They can do both. They can say, for example, that a pre-thingy is a thingy without some property, then say “That is, a pre-thingy satisfies the following axioms: …”